Quantum Energy Teleportation with Electromagnetic Field: Discrete vs. Continuous Variables

0.21 MB·English

by  Masahiro Hotta

Quantum Energy Teleportation
0 with Electromagnetic Field:
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0
Discrete vs. Continuous Variables
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a
J

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Masahiro Hotta

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h
p
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n Department of Physics, Faculty of Science, Tohoku University,
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u Sendai 980-8578, Japan
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[e-mail]
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2 Abstract
v
4 It is well known that usual quantum teleportation protocols can-
7 not transport energy. Recently, new protocols called quantum energy
6
teleportation (QET) have been proposed, which transport energy by
2
. local operations and classical communication with the ground states
8
0 of many-body quantum systems. In this paper, we compare two dif-
9 ferent QET protocols for transporting energy with electromagnetic
0
field. In the first protocol, a 1/2 spin (a qubit) is coupled with the
:
v
quantum fluctuation in the vacuum state and measured in order to
i
X obtain one-bit information about the fluctuation for the teleporta-
r tion. In the second protocol, a harmonic oscillator is coupled with
a
the fluctuation and measured in order to obtain continuous-variable
information about the fluctuation. In the spin protocol, the amount
of teleported energy is suppressed by an exponential damping factor
when the amount of input energy increases. This suppression factor
becomes power damping in the case of the harmonic oscillator proto-
col. Therefore, it is concluded that obtaining more information about
the quantum fluctuation leads to teleporting more energy. This re-
sult suggests a profound relationship between energy and quantum
information.
1 Introduction
In quantum field theory, the concept of negative energy physics has at-
tractedconsiderableattentionforalongtime. Quantuminterference canpro-
duce various states containing regions of negative energy, although the total
energy remains nonnegative [1]. The concept of negative energy plays im-
portant rolesin many fundamental problems ofphysics, including traversable
wormhole [2], cosmic censorship [3], and the second law of thermodynamics
[4]. Inaddition, itsphysical applicationtoquantumopticshasbeendiscussed
[5]. Recently, negative energy physics has yielded a quantum protocol called
quantum energy teleportation (QET) in which energy can be transported
using only local operations and classical communication (LOCC) without
breaking causality and local energy conservation [6]-[8]. QET can be theo-
retically considered in various many-body quantum systems including 1+1
dimensional massless fields [6], spin chains [7] and cold trapped ions [8].
Based on developing measurement technology with sensitive energy resolu-
tion for the systems, the QET effect might be observable in future. It may
be also possible to enhance the amount of teleported energy by preparing
a large number of parallel QET channels, performing a QET protocol for
each channel and accumulating each teleported energy so as to achieve de-
sired amount of total energy. After future experimental verification of QET,
amazing possibility would be open in principle for nano-technology applica-
tion of QET. For example, it may be imagined that, without heat generation
in the intermediate subsystems of the QET channels, energy is transported
in nano-machines at a speed much faster than the evolution speed of excita-
tions of the channels. This technology, if possible, helps future development
of quantum computers in which energy distribution and quantum tasks in
the devices are completed before heat generation in the system. QET is also
expected to provide insights on unsolved problems in gravitational physics.
In fact, a QET process has already been analyzed in black hole physics, and
from the measured information of zero-point oscillation of quantum fields,
it can be regarded as controlled black hole evaporation if we consider the
protocol near the horizon of a large-mass black hole [9].
Energy transportation usually requires physical carriers of energy such
as electric currents and radiation waves. Energy is infused into the gateway
point of a transport channel connected to a distant exit point. Then, energy
1
carriers of the channel excite and propagate to the exit point. At the exit
point, energy is extracted fromthe carriers andharnessed for many purposes.
On the other hand, in the QET protocols, energy can be extracted from the
exit point even if no excited energy carriers arrive at the exit point of the
channel. We locally measure quantum fluctuation around the gateway point
in the ground state of the channel system and announce the measurement
result to the distant exit point with zero energy density, where we can ex-
tract energy from the channel. A key feature is that this measurement result
includes information about the quantum fluctuation of the channel around
this distant point via quantum correlation of the ground state of the chan-
nel system. Therefore, we can infer details about the behavior of a distant
fluctuation from the result of the local measurement. To compensate the ex-
traction of this information, some amount of energy must be infused into the
channel system at the measurement point; this is regarded as input energy to
the gateway point of the channel. By choosing and performing a proper local
operation based on the announced information at the distant point, the local
zero-point oscillation around the distant point can be suppressed relative to
the ground-state one, yielding a negative energy density. During the opera-
tion, respecting local energy conservation, positive amount of surplus energy
is moved from the channel system to external systems. This is regarded as
output teleported energy from the exit point of the channel.
One of the important unresolved problems in QET is the theoretical clar-
ification of the properties in 1+3 dimensions. Protocols in 1+1 dimensions
have already been extensively analyzed in previous studies [6]-[8]. However,
1+3 dimensional models have not yet been analyzed. In addition, all the pro-
tocols proposed thus far adopt quantum measurements for discrete-variable
information. Therefore, it would be interesting to investigate not only a
protocol with discrete-variable information but also one with continuous-
variable information. In this study, we carry out a detailed analysis of two
QET protocols for 1+3 dimensional electromagnetic field in the Coulomb
gauge. Local measurements of quantum fluctuations in the vacuum state
of the field require energy infusion to the field. The infused energy is dif-
fused to spatial infinity at the velocity of light and the state of the field soon
becomes a local vacuum with zero energy around the measurement area. Ob-
viously, this escaped energy cannot be taken back to the measurement area
by local operations around this area if we do not know the measurement
result of the fluctuation. However, if the measurement result is available,
we can effectively take back a part of this energy to the measurement area
2
by applying the QET mechanism. By carrying out a local unitary operation
dependent on the measurement result for the measurement area with zero
energy density, the fluctuation of zero-point oscillation is squeezed and a neg-
ative energy density appears around the area, accompanied by the extraction
of positive energy from the fluctuation to external systems. Needless to say,
without the measurement result, it is impossible to extract energy from the
zero-energy fluctuation. One of the two QET protocols we will consider is
a teleportation in which discrete-variable information about a fluctuation is
obtained using a measurement with a 1/2 spin (a qubit), and the other is
a teleportation in which continuous-variable information is obtained using
a measurement with a harmonic oscillator. The discrete-variable protocol
is a straightforward extension of the protocol for a 1+1 dimensional field
proposed in [6]. The measurements are generalized (POVM) ones that use
probe systems (1/2 spin and harmonic oscillator) strongly interacting with
local electric field fluctuations during a short time. We prove that for a large
energy input, the continuous-variable teleportation is more attractive than
the discrete-variable teleportation. In the discrete-variable case, the amount
of teleported energy is suppressed by an exponential damping factor when
the amount of energy infused by the measurement increases. Meanwhile, this
suppression factor becomes power damping in the continuous-variable case.
Therefore, it is concluded that obtaining more information about the quan-
tum fluctuation leads to teleporting more energy. This result suggests a new
profoundrelationbetween energy andquantum information. So far, relation-
ship between energy and information has been extensively discussed only in
the context of computation energy cost [10], [11], [12]. The QET viewpoint
may shed light on a new relationship between amount of teleported energy
and amount of quantum information about ground-state fluctuations which
would be characterized by various informational indices including mutual in-
formation and entanglement. The explicit analysis about this relationship is
beyond the scope of this paper and will be reported elsewhere.
The remainder of this paper is organized as follows. In section 2, a brief
review of the quantization of the electromagnetic field in the Coulomb gauge
is presented in order to clarify our notations, and the emergence of negative
energy density is explained. In section 3, we discuss a discrete-variable pro-
tocol. In section 4, a continuous-variable protocol is analyzed. In section 5, a
summary and discussions are presented. In this paper, we adopt the natural
unit c = ~ = 1.
3
2 Quantization in Coulomb Gauge
We present a short review of quantization of the electromagnetic field in
theCoulomb gaugeinorder toclarifythe notationsused forlater discussions.
The gauge is defined by
A = 0, divA = 0
0
for the gauge field A =(A ,A). Then, the equation of motion of the Heisen-
µ 0
berg operator of the gauge field Aˆ(t,x) is reduced to the massless Klein-
Gordon equation given by
∂2 2 Aˆ = 0.
t −∇
(cid:2) (cid:3)
The solution can be expanded in terms of plain-wave modes as follows.
d3k
Aˆ(t,x) = e (k)aˆhei(k·x−|k|t) +e∗ (k)aˆh†e−i(k·x−|k|t) ,
h k h k
Z (2π)32|k| hX=1,2h i
q
where aˆh† aˆh is a creation (annihilation) operator of the photon with
k k
momentum k and polarization h satisfying
(cid:0) (cid:1)
aˆh, aˆh′† = δ δ(k k′),
k k′ hh′ −
h i
and e (k) is a polarization vector satisfying
h
e (k)∗ e (k) = δ ,
h h′ hh′
·
k e (k) = 0.
h
·
In this study, because we take a sum of two polarization contributions to
obtain the final results, the reality condition can be imposed on e (k) for
h
simplicity such that e∗ (k) = e (k). In addition, e (k) satisfies the com-
h h h
pleteness relation as
k k
ea(k)eb (k) = δ a b.
h h ab − k2
h
X
The energy density operator of the field is defined by
4
1 2
εˆ(x) = : Eˆ (x)2 + Aˆ (x) :,
2 ∇×
(cid:18) (cid:19)
(cid:16) (cid:17)
where Eˆ (x) is the electric field operator and :: denotes the standard normal
order with respect to aˆh and aˆh†. The Hamiltonian is given by the spatial
k k
integration of the energy density as follows.
Hˆ = εˆ(x)d3x.
Z
It is a well-known fact that the Hamiltonian (total energy of the field) is a
nonnegative operator. The vacuum state 0 is the eigenstate with the lowest
| i
eigenvalue zero of Hˆ as Hˆ 0 = 0. The expectation value of energy density
| i
vanishes for the vacuum state as
0 εˆ(x) 0 = 0. (1)
h | | i
In the later discussion, coherent states are used often. Therefore, we
present a summary of the related properties of the coherent states. A dis-
placement operator generating a coherent state from 0 is given by
| i
Uˆ (p,q) = exp i p(x) Aˆ(x) q(x) Eˆ(x) d3x ,
· − ·
(cid:20) Z h i (cid:21)
where p(x) and q(x) are real vector functions satisfying the conditions of
this gauge as
p(x) = 0,
∇·
q(x) = 0.
∇·
By using the commutation relation between the gauge field and the electric
field at time t = 0 given by
∂ ∂
Aˆ (x), Eˆ (y) = i δ a b δ(x y),
a b ab − 2 −
h i (cid:18) ∇ (cid:19)
it is easily verified that Uˆ (p,q) displaces Aˆ(x) and Eˆ(x) as
Uˆ†(p,q)Eˆ(x)Uˆ (p,q) = Eˆ(x)+p(x), (2)
Uˆ†(p,q)Aˆ(x)Uˆ (p,q) = Aˆ(x)+q(x). (3)
5
In addition, we are able to prove a product formula as
Uˆ (p ,q )Uˆ (p ,q )
1 1 2 2
i
= exp (p q q p )d3x Uˆ (p +p ,q +q ).
2 1· 2− 1· 2 1 2 1 2
(cid:20) Z (cid:21)
This implies that the set of Uˆ (p,q) forms a unitary ray representation of
the displacement group of the field. Coherent states generated by Uˆ (p,q)
are defined by
(p,q) = Uˆ (p,q) 0 . (4)
| i | i
By using the Fourier transformation of p(x) and q(x) defined by
P(k) = p(x)e−ik·xd3x,
Z
Q(k) = q(x)e−ik·xd3x,
Z
the coherent states are explicitly written in terms of the creation operator
aˆh† as follows.
k
(p,q)
| i
1 d3k
= exp P(k) i k Q(k) 2
−2 (2π)32 k | − | | |
(cid:20) Z | | (cid:21)
d3k
exp i e (k) (P(k) i k Q(k))ah† 0 .
×  h · − | | k | i
Z (2π)32 k h
| | X
 q 
From the above expression, it is easy to prove that (p,q) is an eigenstate
| i
of the annihilation operator aˆh such that
k
i
aˆh (p,q) = e (k) (P(k) i k Q(k)) (p,q) . (5)
k h
| i · − | | | i
(2π)32 k
| |
q
The inner product of two coherent states is explicitly calculated as
(p ,q ) (p ,q )
h 1 1 | 2 2 i
= e2i R(p1·q2−q1·p2)d3x
e−12R (2πd)33k2|k||P1(k)−P2(k)−i|k|(Q1(k)−Q2(k))|2. (6)
×
6
Next, we examine the emergence of a region with negative energy density
in this standard theory. As a simple example [4], let us consider a super-
position state Ψ of the vacuum state 0 and a two-photon state 2 such
| i | i | i
that
Ψ = cosθ 0 +eiδsinθ 2 ,
| i | i | i
where θ and δ are real parameters with θ [0,π] and δ [0,2π]. Generally,
∈ ∈
an off-diagonal element of the energy density 0 εˆ(x) 2 does not vanish
h | | i
for a fixed point x. This is because εˆ(x) includes a non-vanishing term
proportional to aˆhaˆh′. This fact indicates the emergence of negativity of the
k k′
energy density as follows. The expectation value of the energy density for
the state Ψ is calculated as
| i
Ψ εˆ(x) Ψ
h | | i
= 2cosθsinθ(cosδRe 0 εˆ(x) 2 sinδIm 0 εˆ(x) 2 )
h | | i− h | | i
+sin2θ 2 εˆ(x) 2 .
h | | i
In this result, let us set the parameters θ and δ so as to satisfy
2 εˆ(x) 2
cosθ = h | | i ,
2 εˆ(x) 2 2 +4 0 εˆ(x) 2 2
h | | i |h | | i|
q 2 0 εˆ(x) 2
sinθ = |h | | i| ,
2 εˆ(x) 2 2 +4 0 εˆ(x) 2 2
h | | i |h | | i|
qIm 0 εˆ(x) 2
tanδ = h | | i.
−Re 0 εˆ(x) 2
h | | i
Then, Ψ εˆ(x) Ψ is evaluated as a negative value as follows.
h | | i
1
Ψ εˆ(x) Ψ = 2 εˆ(x) 2 2 +4 0 εˆ(x) 2 2 2 εˆ(x) 2 < 0.
h | | i −2 h | | i |h | | i| −h | | i
(cid:20)q (cid:21)
Therefore, the emergence of negative-energy regions is not a peculiar phe-
nomenon in quantum field theory. Quantum interference in the superposi-
tion of photon number eigenstates yields negative values. It should be re-
emphasizedthatdespitetheexistence ofregionswithnegativeenergydensity,
7
the expectation values of Hˆ remain nonnegative. This implies that there ex-
ist regions with a sufficient amount of positive energy so as to make the total
energy nonnegative. As described in sections 3 and 4, this negative energy
plays a crucial role in the QET protocols.
3 Discrete-Variable Teleportation
Our protocol for QET with a 1/2 spin probe is a straightforward extension
of that in [6] and comprises the following three steps.
(1) At time t = 0, the spin probe is strongly coupled with the vacuum
fluctuation of the electric field within a finite compact region V during a
m
very short time. In this process, some information about the fluctuation is
imprinted into the probe. Positive energy is inevitably infused into the field
during the measurement process, as seen later. The amount of energy is
denoted by E . When this energy is infused, positive-energy wave packets
m
of the field are generated and these propagate to spatial infinity with the
velocity of light.
(2) After switching off the interaction, projective measurement of the z-
component of the spin is carried out. If the up or down state is observed,
we assign s = + or , respectively, to the measurement result. This implies
−
that we obtains one-bit information about the field fluctuation via the probe
measurement.
(3) At time t = T, it is assumed that the measurement has finished and
the wave packets have already escaped from the region. Hence, the energy
density inthe regionV isexactly zero. Then, alocal displacement operation
m
is carried out depending on s within V . Even though we have zero energy
m
in V , positive energy is extracted from the field fluctuation during the local
m
operation, generating negative-energy wave packets of the field. The amount
ofnegativeenergyofthewavepackets isdenotedbyE (= E ). Therefore,
o o
−| |
the amount of energy extracted energy from the field is given by + E .
o
| |
In step (1), the measurement interaction between the electric field and
the spin probe is given by
8
Hˆ (t) = g(t)σˆ Gˆ, (7)
m z
where σˆ is the z-component of the Pauli matrices of the spin probe; g(t), a
z
time-dependent real coupling constant; and Gˆ, a Hermitian operator defined
by
π
Gˆ = a (x) Eˆ(x)d3x. (8)
m
4 − ·
Z
Here, a (x) is a real vector function with a support equal to V satisfying
m m
a (x) = 0. In addition, by taking a short-time limit for switching the
m
∇ ·
interaction, we set
g(t) = δ(t 0), (9)
−
The initial state of the spin probe is assumed to be the up state + of the
x
| i
x-component of the spin given by
1 1
+ = .
| xi √2 1
(cid:20) (cid:21)
In step (2), the measurement operators Mˆ [13] are defined by
±
tm
Mˆ = Texp i Hˆ (t)dt + , (10)
± x m x
h± | − | i
(cid:20) Z0 (cid:21)
where is the down state of the x-component of the spin given by
x
|− i
i 1
= − .
|−xi √2 1
(cid:20) − (cid:21)
By using Eq. (9), Mˆ are computed as
±
Mˆ = cosGˆ, (11)
+
Mˆ = sinGˆ. (12)
−
Using Eq. (4), the post-measurement states of the field obtaining s = are
±
calculated as
9